The aircraft routing problem is one of the most studied problems of operations research applied to aircraft management. It involves assigning flights to aircraft while ensuring regular visits to maintenance bases. This paper examines two aspects of the problem. First, we explore the relationship between periodic instances, where flights are the same every day, and periodic solutions. The literature has implicitly assumed-without discussion-that periodic instances necessitate periodic solutions, and even periodic solutions in a stronger form, where every two airplanes perform either the exact same cyclic sequence of flights, or completely disjoint cyclic sequences. However, enforcing such periodicity may eliminate feasible solutions. We prove that, when regular maintenance is required at most every four days, there always exist periodic solutions of this form. Second, we consider the computational hardness of the problem. Even if many papers in this area refer to the NP-hardness of the aircraft routing problem, such a result is only available in the literature for periodic instances. We establish its NP-hardness for a non-periodic version. Polynomiality of a special but natural case is also proven.

翻译：飞机航线规划问题是运筹学应用于飞机管理领域中最受关注的问题之一，其核心在于为飞机分配航班任务，同时确保飞机定期返回维修基地进行检修。本文从两个维度探讨该问题。首先，我们研究了周期性实例（即每日航班安排相同）与周期性解之间的关系。现有文献在未加讨论的情况下隐含假设周期性实例必然对应周期性解，甚至是一种更强形式的周期性解，即任意两架飞机要么执行完全相同的循环航班序列，要么执行完全不相交的循环序列。然而，强制实施此类周期性可能会排除可行的解决方案。我们证明，当常规维修要求最多每四天进行一次时，始终存在这种形式的周期性解。其次，我们考察了该问题的计算复杂性。尽管该领域许多论文提及飞机航线规划问题的NP难性，但文献中仅针对周期性实例给出了相关结论。我们首次证明了非周期性版本同样具有NP难性。同时，本文还证明了一个特殊但自然情况下的多项式时间可解性。